Legendre Differential Equation -- from Wolfram MathWorld

The Legendre differential equation is the second-order ordinary differential equation

(1)

which can be rewritten

(2)

The above form is a special case of the so-called "associated Legendre differential equation" corresponding to the case . The Legendre differential equation has regular singular points at , 1, and .

If the variable is replaced by , then the Legendre differential equation becomes

(3)

derived below for the associated () case.

Since the Legendre differential equation is a second-order ordinary differential equation, it has two linearly independent solutions. A solution which is regular at finite points is called a Legendre function of the first kind, while a solution which is singular at is called a Legendre function of the second kind. If is an integer, the function of the first kind reduces to a polynomial known as the Legendre polynomial.

The Legendre differential equation can be solved using the Frobenius method by making a series expansion with ,

			(4)
			(5)
			(6)

Plugging in,

(1-x^2)sum_(n=0)^inftyn(n-1)a_nx^(n-2)-2xsum_(n=0)^inftyna_nx^(n-1)+l(l+1)sum_(n=0)^inftya_nx^n=0

(7)

	(8)
	(9)
	(10)
	(11)
	(12)
	(13)

$sum_(n=0)^infty{(n+1)(n+2)a_(n+2)+[-n(n-1)-2n+l(l+1)]a_n}=0,$

(14)

so each term must vanish and

(15)

			(16)
			(17)

Therefore,

			(18)
			(19)
			(20)
			(21)
			(22)

so the even solution is

y_1(x)=1+sum_(n=1)^infty(-1)^n([(l-2n+2)...(l-2)l][(l+1)(l+3)...(l+2n-1)])/((2n)!)x^(2n).

(23)

Similarly, the odd solution is

y_2(x)=x+sum_(n=1)^infty(-1)^n([(l-2n+1)...(l-3)(l-1)][(l+2)(l+4)...(l+2n)])/((2n+1)!)x^(2n+1).

(24)

If is an even integer, the series reduces to a polynomial of degree with only even powers of and the series diverges. If is an odd integer, the series reduces to a polynomial of degree with only odd powers of and the series diverges. The general solution for an integer is then given by the Legendre polynomials